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On bounded continuous solutions of the archetypical functional equation with rescaling | | Leonid V. Bogachev
; Gregory Derfel
; Stanislav A. Molchanov
; | | Date: |
19 Sep 2014 | | Abstract: | We study the "archetypical" functional equation $y(x)=iint_{mathbb{R}^2}
y(a(x-b)),mu(mathrm{d}a,mathrm{d}b)$ ($xinmathbb{R}$), where $mu$ is a
probability measure; equivalently, $y(x)=mathbb{E}{y(alpha(x-eta))}$,
where $mathbb{E}$ denotes expectation and $(alpha,eta)$ is random with
distribution $mu$. Particular cases include: (i) $y(x)=sum_{i} p_{i},
y(a_i(x-b_i))$ and (ii) $y’(x)+y(x) =sum_{i} p_{i},y(a_i(x-c_i))$ (pantograph
equation), both subject to the balance condition $sum_{i} p_{i}=1$
(${p_{i}>0}$). Solutions $y(x)$ admit interpretation as harmonic functions of
an associated Markov chain $(X_n)$ with jumps of the form
$x
ightsquigarrowalpha(x-eta)$. The paper concerns Liouville-type results
asserting that any bounded continuous harmonic function is constant. The
problem is essentially governed by the value
$K:=iint_{mathbb{R}^2}ln|a|,mu(mathrm{d}a,mathrm{d}b)=mathbb{E}{ln
|alpha|}$. In the critical case $K=0$, we prove a Liouville theorem subject
to the uniform continuity of $y(x)$. The latter is guaranteed under a mild
regularity assumption on the distribution of $eta$, which is satisfied for a
large class of examples including the pantograph equation (ii). Functional
equation (i) is considered with $a_i=q^{m_i}$ ($q>1$, $m_iinmathbb{Z}$),
whereby a Liouville theorem for $K=0$ can be established without the uniform
continuity assumption. Our results also include a generalization of the
classical Choquet--Deny theorem to the case $|alpha|equiv1$, and a surprising
Liouville theorem in the resonance case $alpha(c-eta)equiv c$. The proofs
systematically employ Doob’s Optional Stopping Theorem (with suitably chosen
stopping times) applied to the martingale $y(X_n)$. | | Source: | arXiv, 1409.5648 | | Services: | Forum | Review | PDF | Favorites | |
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